Abstract
Background We have run the simplified Naomi model using a range of inference methods: TMB, aghq, adam and tmbstan.
Task In this report, we compare the accuracy of the posterior distributions obtained from these inference methods using histograms, Kolmogorov-Smirnov tests, maximum mean discrepancy and Pareto-smoothed importance sampling.
We compare the inference results from TMB, aghq, adam, and tmbstan.
Import these inference results as follows:
tmb <- readRDS("depends/tmb.rds")
aghq <- readRDS("depends/aghq.rds")
adam <- readRDS("depends/adam.rds")
tmbstan <- readRDS("depends/tmbstan.rds")
depends <- yaml::read_yaml("orderly.yml")$depends
For more information about the conditions under which these results were generated, see:
TMBdependency_details <- function(i) {
report_name <- names(depends[[i]])
print(paste0("Inference results obtained from ", report_name, " with the query ", depends[[i]][[report_name]]$id))
report_id <- orderly::orderly_search(query = depends[[i]][[report_name]]$id, report_name)
print(paste0("Obtained report had ID ", report_id, " and was run with the following parameters:"))
print(orderly::orderly_info(report_id, report_name)$parameters)
}
dependency_details(1)
## [1] "Inference results obtained from naomi-simple_fit with the query latest(parameter:tmb == TRUE)"
## [1] "Obtained report had ID 20230221-103645-22c5613b and was run with the following parameters:"
## $tmb
## [1] TRUE
##
## $aghq
## [1] FALSE
##
## $k
## [1] 2
##
## $ndConstruction
## [1] "product"
##
## $tmbstan
## [1] FALSE
##
## $niter
## [1] 1000
##
## $nthin
## [1] 1
##
## $adam
## [1] FALSE
##
## $nsample
## [1] 5000
##
## $area_level
## [1] 4
aghqdependency_details(2)
## [1] "Inference results obtained from naomi-simple_fit with the query latest(parameter:aghq == TRUE && parameter:k == 1)"
## [1] "Obtained report had ID 20230221-104545-c5e8a287 and was run with the following parameters:"
## $aghq
## [1] TRUE
##
## $k
## [1] 1
##
## $ndConstruction
## [1] "product"
##
## $tmb
## [1] FALSE
##
## $tmbstan
## [1] FALSE
##
## $niter
## [1] 1000
##
## $nthin
## [1] 1
##
## $adam
## [1] FALSE
##
## $nsample
## [1] 5000
##
## $area_level
## [1] 4
adamdependency_details(3)
## [1] "Inference results obtained from naomi-simple_fit with the query latest(parameter:adam == TRUE)"
## [1] "Obtained report had ID 20230221-105031-95fa53bb and was run with the following parameters:"
## $adam
## [1] TRUE
##
## $tmb
## [1] FALSE
##
## $aghq
## [1] FALSE
##
## $k
## [1] 2
##
## $ndConstruction
## [1] "product"
##
## $tmbstan
## [1] FALSE
##
## $niter
## [1] 1000
##
## $nthin
## [1] 1
##
## $nsample
## [1] 5000
##
## $area_level
## [1] 4
tmbstandependency_details(4)
## [1] "Inference results obtained from naomi-simple_fit with the query latest(parameter:tmbstan == TRUE)"
## [1] "Obtained report had ID 20230114-142916-efabd65d and was run with the following parameters:"
## $tmbstan
## [1] TRUE
##
## $niter
## [1] 20000
##
## $nthin
## [1] 20
##
## $tmb
## [1] FALSE
##
## $aghq
## [1] FALSE
##
## $k
## [1] 2
##
## $nsample
## [1] 1000
data.frame(
"TMB" = tmb$time,
"aghq" = aghq$time,
"adam" = adam$time,
"tmbstan" = tmbstan$time
)
adam <- adam$adam
We create histograms and empirical cumulative distribution function (ECDF) difference plots of the samples from each method. All of the possible parameter names are as follows:
pars <- unique(names(tmb$fit$obj$env$par))
pars
## [1] "beta_rho" "beta_alpha" "beta_lambda" "beta_anc_rho" "beta_anc_alpha"
## [6] "logit_phi_rho_x" "log_sigma_rho_x" "logit_phi_rho_xs" "log_sigma_rho_xs" "logit_phi_rho_a"
## [11] "log_sigma_rho_a" "logit_phi_rho_as" "log_sigma_rho_as" "log_sigma_rho_xa" "u_rho_x"
## [16] "us_rho_x" "u_rho_xs" "us_rho_xs" "u_rho_a" "u_rho_as"
## [21] "logit_phi_alpha_x" "log_sigma_alpha_x" "logit_phi_alpha_xs" "log_sigma_alpha_xs" "logit_phi_alpha_a"
## [26] "log_sigma_alpha_a" "logit_phi_alpha_as" "log_sigma_alpha_as" "log_sigma_alpha_xa" "u_alpha_x"
## [31] "us_alpha_x" "u_alpha_xs" "us_alpha_xs" "u_alpha_a" "u_alpha_as"
## [36] "u_alpha_xa" "OmegaT_raw" "log_betaT" "log_sigma_lambda_x" "ui_lambda_x"
## [41] "log_sigma_ancrho_x" "log_sigma_ancalpha_x" "ui_anc_rho_x" "ui_anc_alpha_x" "log_sigma_or_gamma"
## [46] "log_or_gamma"
We will produce plots about the following subset of them. There is no particular reason to choose this subset rather than other, it’s quite arbitrary.
pars_eval <- pars %in% c("beta_rho", "beta_alpha", "beta_lambda", "beta_anc_rho", "beta_anc_alpha", "u_rho_x", "us_rho_x", "u_rho_xs")
names(pars_eval) <- pars
The beta_rho parameter is used in other reports as a check, so create the dataframe to save as artefact here:
beta_rho <- histogram_and_ecdf("beta_rho", i = 1, return_df = TRUE)
saveRDS(beta_rho$df, "beta_rho.rds")
beta_rhohistogram_and_ecdf_helper <- function(par) lapply(1:sum(names(tmb$fit$obj$env$par) == par), histogram_and_ecdf, par = par)
histogram_and_ecdf_helper("beta_rho")
## [[1]]
##
## [[2]]
beta_alphahistogram_and_ecdf_helper("beta_alpha")
## [[1]]
##
## [[2]]
beta_lambdahistogram_and_ecdf_helper("beta_lambda")
## [[1]]
##
## [[2]]
beta_anc_rhohistogram_and_ecdf("beta_anc_rho")
beta_anc_alphahistogram_and_ecdf("beta_anc_alpha")
logitlapply(pars[stringr::str_starts(pars, "logit")], histogram_and_ecdf)
log_sigmalapply(pars[stringr::str_starts(pars, "log_sigma")], histogram_and_ecdf)
u_rho_xhistogram_and_ecdf_helper("u_rho_x")
## [[1]]
##
## [[2]]
##
## [[3]]
##
## [[4]]
##
## [[5]]
##
## [[6]]
##
## [[7]]
##
## [[8]]
##
## [[9]]
##
## [[10]]
##
## [[11]]
##
## [[12]]
##
## [[13]]
##
## [[14]]
##
## [[15]]
##
## [[16]]
##
## [[17]]
##
## [[18]]
##
## [[19]]
##
## [[20]]
##
## [[21]]
##
## [[22]]
##
## [[23]]
##
## [[24]]
##
## [[25]]
##
## [[26]]
##
## [[27]]
##
## [[28]]
##
## [[29]]
##
## [[30]]
##
## [[31]]
##
## [[32]]
u_rho_xhistogram_and_ecdf_helper("u_rho_x")
## [[1]]
##
## [[2]]
##
## [[3]]
##
## [[4]]
##
## [[5]]
##
## [[6]]
##
## [[7]]
##
## [[8]]
##
## [[9]]
##
## [[10]]
##
## [[11]]
##
## [[12]]
##
## [[13]]
##
## [[14]]
##
## [[15]]
##
## [[16]]
##
## [[17]]
##
## [[18]]
##
## [[19]]
##
## [[20]]
##
## [[21]]
##
## [[22]]
##
## [[23]]
##
## [[24]]
##
## [[25]]
##
## [[26]]
##
## [[27]]
##
## [[28]]
##
## [[29]]
##
## [[30]]
##
## [[31]]
##
## [[32]]
us_rho_xhistogram_and_ecdf_helper("us_rho_x")
## [[1]]
##
## [[2]]
##
## [[3]]
##
## [[4]]
##
## [[5]]
##
## [[6]]
##
## [[7]]
##
## [[8]]
##
## [[9]]
##
## [[10]]
##
## [[11]]
##
## [[12]]
##
## [[13]]
##
## [[14]]
##
## [[15]]
##
## [[16]]
##
## [[17]]
##
## [[18]]
##
## [[19]]
##
## [[20]]
##
## [[21]]
##
## [[22]]
##
## [[23]]
##
## [[24]]
##
## [[25]]
##
## [[26]]
##
## [[27]]
##
## [[28]]
##
## [[29]]
##
## [[30]]
##
## [[31]]
##
## [[32]]
u_rho_xshistogram_and_ecdf_helper("u_rho_xs")
## [[1]]
##
## [[2]]
##
## [[3]]
##
## [[4]]
##
## [[5]]
##
## [[6]]
##
## [[7]]
##
## [[8]]
##
## [[9]]
##
## [[10]]
##
## [[11]]
##
## [[12]]
##
## [[13]]
##
## [[14]]
##
## [[15]]
##
## [[16]]
##
## [[17]]
##
## [[18]]
##
## [[19]]
##
## [[20]]
##
## [[21]]
##
## [[22]]
##
## [[23]]
##
## [[24]]
##
## [[25]]
##
## [[26]]
##
## [[27]]
##
## [[28]]
##
## [[29]]
##
## [[30]]
##
## [[31]]
##
## [[32]]
us_rho_xshistogram_and_ecdf_helper("us_rho_xs")
u_rho_ashistogram_and_ecdf_helper("u_rho_as")
u_alpha_xhistogram_and_ecdf_helper("u_alpha_x")
us_alpha_xhistogram_and_ecdf_helper("us_alpha_x")
u_alpha_xshistogram_and_ecdf_helper("u_alpha_xs")
us_alpha_xshistogram_and_ecdf_helper("us_alpha_xs")
## [[1]]
##
## [[2]]
##
## [[3]]
##
## [[4]]
##
## [[5]]
##
## [[6]]
##
## [[7]]
##
## [[8]]
##
## [[9]]
##
## [[10]]
##
## [[11]]
##
## [[12]]
##
## [[13]]
##
## [[14]]
##
## [[15]]
##
## [[16]]
##
## [[17]]
##
## [[18]]
##
## [[19]]
##
## [[20]]
##
## [[21]]
##
## [[22]]
##
## [[23]]
##
## [[24]]
##
## [[25]]
##
## [[26]]
##
## [[27]]
##
## [[28]]
##
## [[29]]
##
## [[30]]
##
## [[31]]
##
## [[32]]
u_alpha_ahistogram_and_ecdf_helper("u_alpha_a")
u_alpha_ashistogram_and_ecdf_helper("u_alpha_as")
u_alpha_xahistogram_and_ecdf_helper("u_alpha_xa")
ui_lambda_xhistogram_and_ecdf_helper("ui_lambda_x")
ui_anc_rho_xhistogram_and_ecdf_helper("ui_anc_rho_x")
ui_anc_alpha_xhistogram_and_ecdf_helper("ui_anc_alpha_x")
log_or_gammahistogram_and_ecdf_helper("log_or_gamma")
ks_helper <- function(par, sw = FALSE, ...) {
to_ks_df(par, starts_with = sw) %>% ks_plot(par, ...)
}
betaks_helper("beta", sw = TRUE)
## Warning: Removed 76 rows containing missing values (`geom_tile()`).
ks_helper("beta", sw = TRUE, method1 = "TMB", method2 = "adam")
## Warning: Removed 76 rows containing missing values (`geom_tile()`).
logitks_helper("logit", sw = TRUE)
## Warning: Removed 76 rows containing missing values (`geom_tile()`).
ks_helper("logit", sw = TRUE, method1 = "TMB", method2 = "adam")
## Warning: Removed 76 rows containing missing values (`geom_tile()`).
log_sigmaks_helper("log_sigma", sw = TRUE)
## Warning: Removed 76 rows containing missing values (`geom_tile()`).
ks_helper("log_sigma", sw = TRUE , method1 = "TMB", method2 = "adam")
## Warning: Removed 76 rows containing missing values (`geom_tile()`).
u_rho_xks_helper("u_rho_x")
## Warning: Removed 76 rows containing missing values (`geom_tile()`).
ks_helper("u_rho_x", method1 = "TMB", method2 = "adam")
## Warning: Removed 76 rows containing missing values (`geom_tile()`).
u_rho_xsks_helper("u_rho_xs")
## Warning: Removed 76 rows containing missing values (`geom_tile()`).
ks_helper("u_rho_xs", method1 = "TMB", method2 = "adam")
## Warning: Removed 76 rows containing missing values (`geom_tile()`).
us_rho_xks_helper("us_rho_x")
## Warning: Removed 76 rows containing missing values (`geom_tile()`).
ks_helper("us_rho_x", method1 = "TMB", method2 = "adam")
## Warning: Removed 76 rows containing missing values (`geom_tile()`).
us_rho_xsks_helper("us_rho_xs")
ks_helper("us_rho_xs", method1 = "TMB", method2 = "adam")
u_rho_aks_helper("u_rho_a")
ks_helper("u_rho_a", method1 = "TMB", method2 = "adam")
u_rho_asks_helper("u_rho_as")
ks_helper("u_rho_as", method1 = "TMB", method2 = "adam")
u_alpha_xks_helper("u_alpha_x")
ks_helper("u_alpha_x", method1 = "TMB", method2 = "adam")
u_alpha_xsks_helper("u_alpha_xs")
ks_helper("u_alpha_xs", method1 = "TMB", method2 = "adam")
us_alpha_xks_helper("us_alpha_x")
ks_helper("us_alpha_x", method1 = "TMB", method2 = "adam")
us_alpha_xsks_helper("us_alpha_xs")
ks_helper("us_alpha_xs", method1 = "TMB", method2 = "adam")
u_alpha_aks_helper("u_alpha_a")
ks_helper("u_alpha_a", method1 = "TMB", method2 = "adam")
u_alpha_asks_helper("u_alpha_as")
ks_helper("u_alpha_as", method1 = "TMB", method2 = "adam")
u_alpha_xaks_helper("u_alpha_xa")
ks_helper("u_alpha_xa", method1 = "TMB", method2 = "adam")
ui_anc_rho_xks_helper("ui_anc_rho_x")
ks_helper("ui_anc_rho_x", method1 = "TMB", method2 = "adam")
ui_anc_alpha_xks_helper("ui_anc_alpha_x")
ks_helper("ui_anc_alpha_x", method1 = "TMB", method2 = "adam")
log_or_gammaks_helper("log_or_gamma")
ks_helper("log_or_gamma", method1 = "TMB", method2 = "adam")
options(dplyr.summarise.inform = FALSE)
ks_summary <- lapply(unique(names(tmb$fit$obj$env$par)), function(x) {
to_ks_df(x) %>%
group_by(method, index) %>%
summarise(ks = mean(ks), par = x) %>%
ungroup()
}) %>%
bind_rows() %>%
pivot_wider(names_from = "method", values_from = "ks") %>%
rename(
"Parameter" = "par",
"KS(adam, tmbstan)" = "adam",
"KS(aghq, tmbstan)" = "aghq",
"KS(TMB, tmbstan)" = "TMB",
)
r <- adam$quad$obj$env$random
x_names <- names(adam$quad$obj$env$par[r])
theta_names <- names(adam$quad$obj$env$par[-r])
dict <- data.frame(
Parameter = c(unique(x_names), unique(theta_names)),
Type = c(rep("Latent field", length(unique(x_names))), rep("Hyper", length(unique(theta_names))))
)
ks_summary <- ks_summary %>%
left_join(dict, by = "Parameter")
Values in green are the lowest KS difference for that particular parameter:
ks_summary %>%
gt::gt() %>%
gt::fmt_number(
columns = starts_with("KS"),
decimals = 3
) %>%
gt::tab_style(
style = cell_fill(color = "#009E73"),
locations = cells_body(
columns = `KS(adam, tmbstan)`,
rows = `KS(adam, tmbstan)` < `KS(aghq, tmbstan)` & `KS(adam, tmbstan)` < `KS(TMB, tmbstan)`
)
) %>%
gt::tab_style(
style = cell_fill(color = "#009E73"),
locations = cells_body(
columns = `KS(aghq, tmbstan)`,
rows = `KS(aghq, tmbstan)` < `KS(adam, tmbstan)` & `KS(aghq, tmbstan)` < `KS(TMB, tmbstan)`
)
) %>%
gt::tab_style(
style = cell_fill(color = "#009E73"),
locations = cells_body(
columns = `KS(TMB, tmbstan)`,
rows = `KS(TMB, tmbstan)` < `KS(aghq, tmbstan)` & `KS(TMB, tmbstan)` < `KS(adam, tmbstan)`
)
)
| index | Parameter | KS(adam, tmbstan) | KS(aghq, tmbstan) | KS(TMB, tmbstan) | Type |
|---|---|---|---|---|---|
| 1 | beta_rho | 0.087 | 0.082 | 0.087 | Latent field |
| 2 | beta_rho | 0.086 | 0.085 | 0.089 | Latent field |
| 1 | beta_alpha | 0.055 | 0.058 | 0.071 | Latent field |
| 2 | beta_alpha | 0.132 | 0.130 | 0.131 | Latent field |
| 1 | beta_lambda | 0.048 | 0.086 | 0.079 | Latent field |
| 2 | beta_lambda | 0.016 | 0.024 | 0.016 | Latent field |
| 1 | beta_anc_rho | 0.046 | 0.080 | 0.092 | Latent field |
| 1 | beta_anc_alpha | 0.070 | 0.027 | 0.017 | Latent field |
| 1 | logit_phi_rho_x | 0.562 | 0.562 | 0.133 | Hyper |
| 1 | log_sigma_rho_x | 0.656 | 0.656 | 0.155 | Hyper |
| 1 | logit_phi_rho_xs | 0.510 | 0.510 | 0.104 | Hyper |
| 1 | log_sigma_rho_xs | 0.613 | 0.613 | 0.212 | Hyper |
| 1 | logit_phi_rho_a | 0.537 | 0.537 | 0.058 | Hyper |
| 1 | log_sigma_rho_a | 0.581 | 0.581 | 0.110 | Hyper |
| 1 | logit_phi_rho_as | 0.569 | 0.569 | 0.113 | Hyper |
| 1 | log_sigma_rho_as | 0.589 | 0.589 | 0.142 | Hyper |
| 1 | log_sigma_rho_xa | 0.669 | 0.669 | 0.211 | Hyper |
| 1 | u_rho_x | 0.040 | 0.086 | 0.085 | Latent field |
| 2 | u_rho_x | 0.021 | 0.035 | 0.038 | Latent field |
| 3 | u_rho_x | 0.032 | 0.027 | 0.030 | Latent field |
| 4 | u_rho_x | 0.037 | 0.072 | 0.084 | Latent field |
| 5 | u_rho_x | 0.081 | 0.067 | 0.062 | Latent field |
| 6 | u_rho_x | 0.087 | 0.101 | 0.094 | Latent field |
| 7 | u_rho_x | 0.039 | 0.096 | 0.089 | Latent field |
| 8 | u_rho_x | 0.090 | 0.066 | 0.071 | Latent field |
| 9 | u_rho_x | 0.043 | 0.080 | 0.084 | Latent field |
| 10 | u_rho_x | 0.048 | 0.081 | 0.076 | Latent field |
| 11 | u_rho_x | 0.037 | 0.045 | 0.053 | Latent field |
| 12 | u_rho_x | 0.068 | 0.127 | 0.133 | Latent field |
| 13 | u_rho_x | 0.035 | 0.029 | 0.039 | Latent field |
| 14 | u_rho_x | 0.059 | 0.078 | 0.086 | Latent field |
| 15 | u_rho_x | 0.055 | 0.035 | 0.042 | Latent field |
| 16 | u_rho_x | 0.028 | 0.024 | 0.021 | Latent field |
| 17 | u_rho_x | 0.022 | 0.051 | 0.046 | Latent field |
| 18 | u_rho_x | 0.080 | 0.044 | 0.047 | Latent field |
| 19 | u_rho_x | 0.017 | 0.027 | 0.029 | Latent field |
| 20 | u_rho_x | 0.018 | 0.018 | 0.040 | Latent field |
| 21 | u_rho_x | 0.014 | 0.038 | 0.050 | Latent field |
| 22 | u_rho_x | 0.105 | 0.084 | 0.084 | Latent field |
| 23 | u_rho_x | 0.048 | 0.035 | 0.039 | Latent field |
| 24 | u_rho_x | 0.032 | 0.033 | 0.038 | Latent field |
| 25 | u_rho_x | 0.041 | 0.057 | 0.045 | Latent field |
| 26 | u_rho_x | 0.059 | 0.025 | 0.023 | Latent field |
| 27 | u_rho_x | 0.056 | 0.068 | 0.074 | Latent field |
| 28 | u_rho_x | 0.056 | 0.063 | 0.060 | Latent field |
| 29 | u_rho_x | 0.078 | 0.119 | 0.100 | Latent field |
| 30 | u_rho_x | 0.063 | 0.057 | 0.061 | Latent field |
| 31 | u_rho_x | 0.028 | 0.044 | 0.028 | Latent field |
| 32 | u_rho_x | 0.034 | 0.050 | 0.060 | Latent field |
| 1 | us_rho_x | 0.141 | 0.163 | 0.139 | Latent field |
| 2 | us_rho_x | 0.155 | 0.139 | 0.119 | Latent field |
| 3 | us_rho_x | 0.096 | 0.087 | 0.091 | Latent field |
| 4 | us_rho_x | 0.068 | 0.069 | 0.083 | Latent field |
| 5 | us_rho_x | 0.066 | 0.051 | 0.068 | Latent field |
| 6 | us_rho_x | 0.182 | 0.175 | 0.120 | Latent field |
| 7 | us_rho_x | 0.083 | 0.072 | 0.067 | Latent field |
| 8 | us_rho_x | 0.084 | 0.073 | 0.076 | Latent field |
| 9 | us_rho_x | 0.068 | 0.068 | 0.065 | Latent field |
| 10 | us_rho_x | 0.054 | 0.047 | 0.045 | Latent field |
| 11 | us_rho_x | 0.082 | 0.078 | 0.088 | Latent field |
| 12 | us_rho_x | 0.083 | 0.091 | 0.093 | Latent field |
| 13 | us_rho_x | 0.067 | 0.050 | 0.054 | Latent field |
| 14 | us_rho_x | 0.038 | 0.028 | 0.046 | Latent field |
| 15 | us_rho_x | 0.043 | 0.050 | 0.049 | Latent field |
| 16 | us_rho_x | 0.061 | 0.054 | 0.072 | Latent field |
| 17 | us_rho_x | 0.063 | 0.084 | 0.076 | Latent field |
| 18 | us_rho_x | 0.117 | 0.107 | 0.120 | Latent field |
| 19 | us_rho_x | 0.072 | 0.082 | 0.086 | Latent field |
| 20 | us_rho_x | 0.144 | 0.144 | 0.143 | Latent field |
| 21 | us_rho_x | 0.104 | 0.093 | 0.106 | Latent field |
| 22 | us_rho_x | 0.077 | 0.064 | 0.045 | Latent field |
| 23 | us_rho_x | 0.083 | 0.091 | 0.070 | Latent field |
| 24 | us_rho_x | 0.070 | 0.063 | 0.076 | Latent field |
| 25 | us_rho_x | 0.081 | 0.107 | 0.095 | Latent field |
| 26 | us_rho_x | 0.084 | 0.079 | 0.084 | Latent field |
| 27 | us_rho_x | 0.044 | 0.056 | 0.056 | Latent field |
| 28 | us_rho_x | 0.053 | 0.054 | 0.059 | Latent field |
| 29 | us_rho_x | 0.078 | 0.069 | 0.076 | Latent field |
| 30 | us_rho_x | 0.087 | 0.069 | 0.068 | Latent field |
| 31 | us_rho_x | 0.042 | 0.041 | 0.050 | Latent field |
| 32 | us_rho_x | 0.057 | 0.052 | 0.076 | Latent field |
| 1 | u_rho_xs | 0.067 | 0.083 | 0.087 | Latent field |
| 2 | u_rho_xs | 0.099 | 0.072 | 0.077 | Latent field |
| 3 | u_rho_xs | 0.085 | 0.098 | 0.100 | Latent field |
| 4 | u_rho_xs | 0.070 | 0.091 | 0.095 | Latent field |
| 5 | u_rho_xs | 0.153 | 0.146 | 0.140 | Latent field |
| 6 | u_rho_xs | 0.207 | 0.192 | 0.170 | Latent field |
| 7 | u_rho_xs | 0.106 | 0.109 | 0.102 | Latent field |
| 8 | u_rho_xs | 0.103 | 0.088 | 0.096 | Latent field |
| 9 | u_rho_xs | 0.111 | 0.117 | 0.119 | Latent field |
| 10 | u_rho_xs | 0.113 | 0.111 | 0.107 | Latent field |
| 11 | u_rho_xs | 0.083 | 0.078 | 0.087 | Latent field |
| 12 | u_rho_xs | 0.183 | 0.162 | 0.152 | Latent field |
| 13 | u_rho_xs | 0.099 | 0.102 | 0.112 | Latent field |
| 14 | u_rho_xs | 0.115 | 0.121 | 0.127 | Latent field |
| 15 | u_rho_xs | 0.146 | 0.160 | 0.144 | Latent field |
| 16 | u_rho_xs | 0.101 | 0.093 | 0.088 | Latent field |
| 17 | u_rho_xs | 0.092 | 0.104 | 0.090 | Latent field |
| 18 | u_rho_xs | 0.145 | 0.121 | 0.133 | Latent field |
| 19 | u_rho_xs | 0.109 | 0.120 | 0.126 | Latent field |
| 20 | u_rho_xs | 0.098 | 0.080 | 0.104 | Latent field |
| 21 | u_rho_xs | 0.097 | 0.098 | 0.100 | Latent field |
| 22 | u_rho_xs | 0.202 | 0.192 | 0.173 | Latent field |
| 23 | u_rho_xs | 0.105 | 0.100 | 0.123 | Latent field |
| 24 | u_rho_xs | 0.151 | 0.138 | 0.148 | Latent field |
| 25 | u_rho_xs | 0.076 | 0.090 | 0.086 | Latent field |
| 26 | u_rho_xs | 0.127 | 0.126 | 0.120 | Latent field |
| 27 | u_rho_xs | 0.138 | 0.151 | 0.145 | Latent field |
| 28 | u_rho_xs | 0.143 | 0.135 | 0.141 | Latent field |
| 29 | u_rho_xs | 0.198 | 0.187 | 0.178 | Latent field |
| 30 | u_rho_xs | 0.116 | 0.108 | 0.115 | Latent field |
| 31 | u_rho_xs | 0.108 | 0.097 | 0.102 | Latent field |
| 32 | u_rho_xs | 0.105 | 0.108 | 0.105 | Latent field |
| 1 | us_rho_xs | 0.033 | 0.029 | 0.019 | Latent field |
| 2 | us_rho_xs | 0.026 | 0.023 | 0.020 | Latent field |
| 3 | us_rho_xs | 0.035 | 0.024 | 0.026 | Latent field |
| 4 | us_rho_xs | 0.023 | 0.030 | 0.020 | Latent field |
| 5 | us_rho_xs | 0.043 | 0.031 | 0.027 | Latent field |
| 6 | us_rho_xs | 0.036 | 0.019 | 0.021 | Latent field |
| 7 | us_rho_xs | 0.031 | 0.019 | 0.032 | Latent field |
| 8 | us_rho_xs | 0.049 | 0.040 | 0.038 | Latent field |
| 9 | us_rho_xs | 0.041 | 0.043 | 0.047 | Latent field |
| 10 | us_rho_xs | 0.034 | 0.028 | 0.037 | Latent field |
| 11 | us_rho_xs | 0.041 | 0.033 | 0.036 | Latent field |
| 12 | us_rho_xs | 0.038 | 0.042 | 0.036 | Latent field |
| 13 | us_rho_xs | 0.033 | 0.044 | 0.052 | Latent field |
| 14 | us_rho_xs | 0.039 | 0.030 | 0.044 | Latent field |
| 15 | us_rho_xs | 0.029 | 0.024 | 0.035 | Latent field |
| 16 | us_rho_xs | 0.022 | 0.017 | 0.015 | Latent field |
| 17 | us_rho_xs | 0.040 | 0.032 | 0.037 | Latent field |
| 18 | us_rho_xs | 0.032 | 0.035 | 0.045 | Latent field |
| 19 | us_rho_xs | 0.031 | 0.033 | 0.035 | Latent field |
| 20 | us_rho_xs | 0.045 | 0.032 | 0.054 | Latent field |
| 21 | us_rho_xs | 0.050 | 0.046 | 0.040 | Latent field |
| 22 | us_rho_xs | 0.045 | 0.041 | 0.040 | Latent field |
| 23 | us_rho_xs | 0.034 | 0.031 | 0.037 | Latent field |
| 24 | us_rho_xs | 0.029 | 0.029 | 0.029 | Latent field |
| 25 | us_rho_xs | 0.046 | 0.047 | 0.059 | Latent field |
| 26 | us_rho_xs | 0.039 | 0.035 | 0.040 | Latent field |
| 27 | us_rho_xs | 0.035 | 0.033 | 0.036 | Latent field |
| 28 | us_rho_xs | 0.026 | 0.024 | 0.023 | Latent field |
| 29 | us_rho_xs | 0.032 | 0.024 | 0.031 | Latent field |
| 30 | us_rho_xs | 0.041 | 0.038 | 0.044 | Latent field |
| 31 | us_rho_xs | 0.041 | 0.044 | 0.045 | Latent field |
| 32 | us_rho_xs | 0.037 | 0.036 | 0.037 | Latent field |
| 1 | u_rho_a | 0.087 | 0.080 | 0.085 | Latent field |
| 2 | u_rho_a | 0.087 | 0.084 | 0.086 | Latent field |
| 3 | u_rho_a | 0.084 | 0.083 | 0.086 | Latent field |
| 4 | u_rho_a | 0.083 | 0.082 | 0.085 | Latent field |
| 5 | u_rho_a | 0.086 | 0.082 | 0.088 | Latent field |
| 6 | u_rho_a | 0.086 | 0.082 | 0.086 | Latent field |
| 7 | u_rho_a | 0.087 | 0.083 | 0.087 | Latent field |
| 8 | u_rho_a | 0.089 | 0.083 | 0.088 | Latent field |
| 9 | u_rho_a | 0.088 | 0.082 | 0.087 | Latent field |
| 10 | u_rho_a | 0.086 | 0.082 | 0.086 | Latent field |
| 1 | u_rho_as | 0.092 | 0.086 | 0.083 | Latent field |
| 2 | u_rho_as | 0.094 | 0.079 | 0.082 | Latent field |
| 3 | u_rho_as | 0.094 | 0.085 | 0.089 | Latent field |
| 4 | u_rho_as | 0.092 | 0.082 | 0.087 | Latent field |
| 5 | u_rho_as | 0.086 | 0.086 | 0.088 | Latent field |
| 6 | u_rho_as | 0.091 | 0.082 | 0.085 | Latent field |
| 7 | u_rho_as | 0.088 | 0.086 | 0.088 | Latent field |
| 8 | u_rho_as | 0.095 | 0.083 | 0.089 | Latent field |
| 9 | u_rho_as | 0.092 | 0.084 | 0.089 | Latent field |
| 10 | u_rho_as | 0.085 | 0.077 | 0.084 | Latent field |
| 1 | logit_phi_alpha_x | 0.512 | 0.512 | 0.073 | Hyper |
| 1 | log_sigma_alpha_x | 0.505 | 0.505 | 0.127 | Hyper |
| 1 | logit_phi_alpha_xs | 0.551 | 0.551 | 0.144 | Hyper |
| 1 | log_sigma_alpha_xs | 0.540 | 0.540 | 0.161 | Hyper |
| 1 | logit_phi_alpha_a | 0.534 | 0.534 | 0.089 | Hyper |
| 1 | log_sigma_alpha_a | 0.569 | 0.569 | 0.107 | Hyper |
| 1 | logit_phi_alpha_as | 0.517 | 0.517 | 0.079 | Hyper |
| 1 | log_sigma_alpha_as | 0.514 | 0.514 | 0.102 | Hyper |
| 1 | log_sigma_alpha_xa | 0.627 | 0.627 | 0.137 | Hyper |
| 1 | u_alpha_x | 0.098 | 0.139 | 0.119 | Latent field |
| 2 | u_alpha_x | 0.119 | 0.130 | 0.115 | Latent field |
| 3 | u_alpha_x | 0.113 | 0.108 | 0.093 | Latent field |
| 4 | u_alpha_x | 0.110 | 0.124 | 0.115 | Latent field |
| 5 | u_alpha_x | 0.138 | 0.146 | 0.147 | Latent field |
| 6 | u_alpha_x | 0.091 | 0.091 | 0.088 | Latent field |
| 7 | u_alpha_x | 0.063 | 0.061 | 0.077 | Latent field |
| 8 | u_alpha_x | 0.126 | 0.137 | 0.110 | Latent field |
| 9 | u_alpha_x | 0.147 | 0.146 | 0.145 | Latent field |
| 10 | u_alpha_x | 0.168 | 0.137 | 0.082 | Latent field |
| 11 | u_alpha_x | 0.162 | 0.154 | 0.151 | Latent field |
| 12 | u_alpha_x | 0.100 | 0.074 | 0.093 | Latent field |
| 13 | u_alpha_x | 0.138 | 0.126 | 0.121 | Latent field |
| 14 | u_alpha_x | 0.084 | 0.084 | 0.098 | Latent field |
| 15 | u_alpha_x | 0.110 | 0.082 | 0.092 | Latent field |
| 16 | u_alpha_x | 0.071 | 0.063 | 0.081 | Latent field |
| 17 | u_alpha_x | 0.046 | 0.055 | 0.052 | Latent field |
| 18 | u_alpha_x | 0.106 | 0.094 | 0.063 | Latent field |
| 19 | u_alpha_x | 0.112 | 0.114 | 0.129 | Latent field |
| 20 | u_alpha_x | 0.093 | 0.103 | 0.099 | Latent field |
| 21 | u_alpha_x | 0.161 | 0.161 | 0.089 | Latent field |
| 22 | u_alpha_x | 0.111 | 0.128 | 0.152 | Latent field |
| 23 | u_alpha_x | 0.160 | 0.146 | 0.132 | Latent field |
| 24 | u_alpha_x | 0.147 | 0.149 | 0.163 | Latent field |
| 25 | u_alpha_x | 0.070 | 0.106 | 0.097 | Latent field |
| 26 | u_alpha_x | 0.079 | 0.097 | 0.074 | Latent field |
| 27 | u_alpha_x | 0.131 | 0.143 | 0.131 | Latent field |
| 28 | u_alpha_x | 0.131 | 0.116 | 0.121 | Latent field |
| 29 | u_alpha_x | 0.094 | 0.105 | 0.133 | Latent field |
| 30 | u_alpha_x | 0.120 | 0.119 | 0.132 | Latent field |
| 31 | u_alpha_x | 0.095 | 0.086 | 0.110 | Latent field |
| 32 | u_alpha_x | 0.066 | 0.063 | 0.091 | Latent field |
| 1 | us_alpha_x | 0.049 | 0.078 | 0.065 | Latent field |
| 2 | us_alpha_x | 0.047 | 0.077 | 0.057 | Latent field |
| 3 | us_alpha_x | 0.064 | 0.070 | 0.067 | Latent field |
| 4 | us_alpha_x | 0.035 | 0.043 | 0.040 | Latent field |
| 5 | us_alpha_x | 0.070 | 0.079 | 0.074 | Latent field |
| 6 | us_alpha_x | 0.059 | 0.058 | 0.053 | Latent field |
| 7 | us_alpha_x | 0.027 | 0.034 | 0.045 | Latent field |
| 8 | us_alpha_x | 0.097 | 0.074 | 0.085 | Latent field |
| 9 | us_alpha_x | 0.107 | 0.091 | 0.101 | Latent field |
| 10 | us_alpha_x | 0.117 | 0.097 | 0.109 | Latent field |
| 11 | us_alpha_x | 0.091 | 0.071 | 0.082 | Latent field |
| 12 | us_alpha_x | 0.071 | 0.047 | 0.062 | Latent field |
| 13 | us_alpha_x | 0.098 | 0.070 | 0.085 | Latent field |
| 14 | us_alpha_x | 0.031 | 0.021 | 0.020 | Latent field |
| 15 | us_alpha_x | 0.073 | 0.043 | 0.054 | Latent field |
| 16 | us_alpha_x | 0.032 | 0.040 | 0.018 | Latent field |
| 17 | us_alpha_x | 0.061 | 0.067 | 0.065 | Latent field |
| 18 | us_alpha_x | 0.065 | 0.055 | 0.056 | Latent field |
| 19 | us_alpha_x | 0.055 | 0.064 | 0.077 | Latent field |
| 20 | us_alpha_x | 0.059 | 0.075 | 0.055 | Latent field |
| 21 | us_alpha_x | 0.102 | 0.089 | 0.091 | Latent field |
| 22 | us_alpha_x | 0.080 | 0.103 | 0.103 | Latent field |
| 23 | us_alpha_x | 0.089 | 0.090 | 0.096 | Latent field |
| 24 | us_alpha_x | 0.095 | 0.086 | 0.100 | Latent field |
| 25 | us_alpha_x | 0.066 | 0.073 | 0.074 | Latent field |
| 26 | us_alpha_x | 0.063 | 0.073 | 0.061 | Latent field |
| 27 | us_alpha_x | 0.046 | 0.055 | 0.046 | Latent field |
| 28 | us_alpha_x | 0.047 | 0.044 | 0.053 | Latent field |
| 29 | us_alpha_x | 0.055 | 0.064 | 0.055 | Latent field |
| 30 | us_alpha_x | 0.039 | 0.064 | 0.060 | Latent field |
| 31 | us_alpha_x | 0.034 | 0.036 | 0.046 | Latent field |
| 32 | us_alpha_x | 0.066 | 0.052 | 0.031 | Latent field |
| 1 | u_alpha_xs | 0.064 | 0.074 | 0.059 | Latent field |
| 2 | u_alpha_xs | 0.061 | 0.083 | 0.064 | Latent field |
| 3 | u_alpha_xs | 0.048 | 0.060 | 0.055 | Latent field |
| 4 | u_alpha_xs | 0.099 | 0.102 | 0.099 | Latent field |
| 5 | u_alpha_xs | 0.116 | 0.122 | 0.129 | Latent field |
| 6 | u_alpha_xs | 0.042 | 0.064 | 0.072 | Latent field |
| 7 | u_alpha_xs | 0.049 | 0.048 | 0.056 | Latent field |
| 8 | u_alpha_xs | 0.075 | 0.063 | 0.051 | Latent field |
| 9 | u_alpha_xs | 0.083 | 0.078 | 0.093 | Latent field |
| 10 | u_alpha_xs | 0.105 | 0.123 | 0.121 | Latent field |
| 11 | u_alpha_xs | 0.119 | 0.119 | 0.115 | Latent field |
| 12 | u_alpha_xs | 0.106 | 0.125 | 0.126 | Latent field |
| 13 | u_alpha_xs | 0.078 | 0.098 | 0.091 | Latent field |
| 14 | u_alpha_xs | 0.046 | 0.048 | 0.062 | Latent field |
| 15 | u_alpha_xs | 0.127 | 0.125 | 0.126 | Latent field |
| 16 | u_alpha_xs | 0.153 | 0.149 | 0.080 | Latent field |
| 17 | u_alpha_xs | 0.055 | 0.058 | 0.054 | Latent field |
| 18 | u_alpha_xs | 0.185 | 0.178 | 0.065 | Latent field |
| 19 | u_alpha_xs | 0.079 | 0.057 | 0.059 | Latent field |
| 20 | u_alpha_xs | 0.058 | 0.064 | 0.064 | Latent field |
| 21 | u_alpha_xs | 0.109 | 0.100 | 0.055 | Latent field |
| 22 | u_alpha_xs | 0.198 | 0.169 | 0.128 | Latent field |
| 23 | u_alpha_xs | 0.096 | 0.087 | 0.106 | Latent field |
| 24 | u_alpha_xs | 0.154 | 0.153 | 0.129 | Latent field |
| 25 | u_alpha_xs | 0.142 | 0.127 | 0.124 | Latent field |
| 26 | u_alpha_xs | 0.047 | 0.049 | 0.060 | Latent field |
| 27 | u_alpha_xs | 0.137 | 0.129 | 0.118 | Latent field |
| 28 | u_alpha_xs | 0.085 | 0.086 | 0.084 | Latent field |
| 29 | u_alpha_xs | 0.193 | 0.176 | 0.094 | Latent field |
| 30 | u_alpha_xs | 0.073 | 0.069 | 0.086 | Latent field |
| 31 | u_alpha_xs | 0.114 | 0.104 | 0.105 | Latent field |
| 32 | u_alpha_xs | 0.118 | 0.099 | 0.081 | Latent field |
| 1 | us_alpha_xs | 0.037 | 0.055 | 0.023 | Latent field |
| 2 | us_alpha_xs | 0.037 | 0.041 | 0.029 | Latent field |
| 3 | us_alpha_xs | 0.049 | 0.064 | 0.041 | Latent field |
| 4 | us_alpha_xs | 0.068 | 0.054 | 0.071 | Latent field |
| 5 | us_alpha_xs | 0.090 | 0.089 | 0.099 | Latent field |
| 6 | us_alpha_xs | 0.059 | 0.069 | 0.074 | Latent field |
| 7 | us_alpha_xs | 0.035 | 0.034 | 0.022 | Latent field |
| 8 | us_alpha_xs | 0.110 | 0.116 | 0.101 | Latent field |
| 9 | us_alpha_xs | 0.122 | 0.117 | 0.117 | Latent field |
| 10 | us_alpha_xs | 0.115 | 0.130 | 0.115 | Latent field |
| 11 | us_alpha_xs | 0.119 | 0.120 | 0.128 | Latent field |
| 12 | us_alpha_xs | 0.129 | 0.136 | 0.145 | Latent field |
| 13 | us_alpha_xs | 0.107 | 0.106 | 0.100 | Latent field |
| 14 | us_alpha_xs | 0.093 | 0.086 | 0.090 | Latent field |
| 15 | us_alpha_xs | 0.120 | 0.126 | 0.118 | Latent field |
| 16 | us_alpha_xs | 0.145 | 0.138 | 0.107 | Latent field |
| 17 | us_alpha_xs | 0.047 | 0.048 | 0.054 | Latent field |
| 18 | us_alpha_xs | 0.098 | 0.091 | 0.050 | Latent field |
| 19 | us_alpha_xs | 0.029 | 0.024 | 0.025 | Latent field |
| 20 | us_alpha_xs | 0.035 | 0.039 | 0.024 | Latent field |
| 21 | us_alpha_xs | 0.062 | 0.052 | 0.044 | Latent field |
| 22 | us_alpha_xs | 0.130 | 0.130 | 0.111 | Latent field |
| 23 | us_alpha_xs | 0.106 | 0.107 | 0.098 | Latent field |
| 24 | us_alpha_xs | 0.157 | 0.157 | 0.144 | Latent field |
| 25 | us_alpha_xs | 0.105 | 0.096 | 0.088 | Latent field |
| 26 | us_alpha_xs | 0.154 | 0.131 | 0.119 | Latent field |
| 27 | us_alpha_xs | 0.146 | 0.129 | 0.130 | Latent field |
| 28 | us_alpha_xs | 0.100 | 0.097 | 0.099 | Latent field |
| 29 | us_alpha_xs | 0.160 | 0.150 | 0.128 | Latent field |
| 30 | us_alpha_xs | 0.168 | 0.153 | 0.139 | Latent field |
| 31 | us_alpha_xs | 0.133 | 0.134 | 0.113 | Latent field |
| 32 | us_alpha_xs | 0.113 | 0.120 | 0.113 | Latent field |
| 1 | u_alpha_a | 0.052 | 0.050 | 0.055 | Latent field |
| 2 | u_alpha_a | 0.054 | 0.052 | 0.064 | Latent field |
| 3 | u_alpha_a | 0.057 | 0.058 | 0.061 | Latent field |
| 4 | u_alpha_a | 0.071 | 0.069 | 0.079 | Latent field |
| 5 | u_alpha_a | 0.051 | 0.052 | 0.066 | Latent field |
| 6 | u_alpha_a | 0.051 | 0.046 | 0.054 | Latent field |
| 7 | u_alpha_a | 0.050 | 0.046 | 0.052 | Latent field |
| 8 | u_alpha_a | 0.053 | 0.046 | 0.057 | Latent field |
| 9 | u_alpha_a | 0.050 | 0.049 | 0.061 | Latent field |
| 10 | u_alpha_a | 0.054 | 0.054 | 0.065 | Latent field |
| 11 | u_alpha_a | 0.062 | 0.060 | 0.072 | Latent field |
| 12 | u_alpha_a | 0.060 | 0.066 | 0.080 | Latent field |
| 13 | u_alpha_a | 0.055 | 0.059 | 0.070 | Latent field |
| 1 | u_alpha_as | 0.070 | 0.111 | 0.120 | Latent field |
| 2 | u_alpha_as | 0.099 | 0.122 | 0.130 | Latent field |
| 3 | u_alpha_as | 0.142 | 0.139 | 0.140 | Latent field |
| 4 | u_alpha_as | 0.145 | 0.132 | 0.138 | Latent field |
| 5 | u_alpha_as | 0.148 | 0.129 | 0.140 | Latent field |
| 6 | u_alpha_as | 0.137 | 0.109 | 0.121 | Latent field |
| 7 | u_alpha_as | 0.102 | 0.082 | 0.100 | Latent field |
| 8 | u_alpha_as | 0.089 | 0.069 | 0.092 | Latent field |
| 9 | u_alpha_as | 0.064 | 0.054 | 0.059 | Latent field |
| 10 | u_alpha_as | 0.051 | 0.043 | 0.051 | Latent field |
| 1 | u_alpha_xa | 0.026 | 0.022 | 0.021 | Latent field |
| 2 | u_alpha_xa | 0.037 | 0.023 | 0.024 | Latent field |
| 3 | u_alpha_xa | 0.020 | 0.025 | 0.020 | Latent field |
| 4 | u_alpha_xa | 0.038 | 0.056 | 0.060 | Latent field |
| 5 | u_alpha_xa | 0.107 | 0.111 | 0.110 | Latent field |
| 6 | u_alpha_xa | 0.087 | 0.065 | 0.065 | Latent field |
| 7 | u_alpha_xa | 0.035 | 0.035 | 0.035 | Latent field |
| 8 | u_alpha_xa | 0.055 | 0.048 | 0.061 | Latent field |
| 9 | u_alpha_xa | 0.043 | 0.046 | 0.052 | Latent field |
| 10 | u_alpha_xa | 0.085 | 0.107 | 0.098 | Latent field |
| 11 | u_alpha_xa | 0.097 | 0.106 | 0.122 | Latent field |
| 12 | u_alpha_xa | 0.116 | 0.111 | 0.110 | Latent field |
| 13 | u_alpha_xa | 0.058 | 0.076 | 0.070 | Latent field |
| 14 | u_alpha_xa | 0.064 | 0.062 | 0.048 | Latent field |
| 15 | u_alpha_xa | 0.044 | 0.041 | 0.058 | Latent field |
| 16 | u_alpha_xa | 0.079 | 0.085 | 0.066 | Latent field |
| 17 | u_alpha_xa | 0.029 | 0.047 | 0.050 | Latent field |
| 18 | u_alpha_xa | 0.055 | 0.057 | 0.044 | Latent field |
| 19 | u_alpha_xa | 0.067 | 0.070 | 0.061 | Latent field |
| 20 | u_alpha_xa | 0.026 | 0.036 | 0.023 | Latent field |
| 21 | u_alpha_xa | 0.062 | 0.069 | 0.044 | Latent field |
| 22 | u_alpha_xa | 0.215 | 0.201 | 0.179 | Latent field |
| 23 | u_alpha_xa | 0.042 | 0.053 | 0.041 | Latent field |
| 24 | u_alpha_xa | 0.083 | 0.070 | 0.065 | Latent field |
| 25 | u_alpha_xa | 0.091 | 0.080 | 0.074 | Latent field |
| 26 | u_alpha_xa | 0.024 | 0.042 | 0.047 | Latent field |
| 27 | u_alpha_xa | 0.079 | 0.088 | 0.078 | Latent field |
| 28 | u_alpha_xa | 0.033 | 0.027 | 0.020 | Latent field |
| 29 | u_alpha_xa | 0.030 | 0.038 | 0.036 | Latent field |
| 30 | u_alpha_xa | 0.040 | 0.043 | 0.047 | Latent field |
| 31 | u_alpha_xa | 0.028 | 0.038 | 0.030 | Latent field |
| 32 | u_alpha_xa | 0.021 | 0.035 | 0.025 | Latent field |
| 1 | OmegaT_raw | 0.518 | 0.518 | 0.025 | Hyper |
| 1 | log_betaT | 0.689 | 0.689 | 0.245 | Hyper |
| 1 | log_sigma_lambda_x | 0.736 | 0.736 | 0.258 | Hyper |
| 1 | ui_lambda_x | 0.158 | 0.150 | 0.147 | Latent field |
| 2 | ui_lambda_x | 0.132 | 0.137 | 0.144 | Latent field |
| 3 | ui_lambda_x | 0.166 | 0.153 | 0.169 | Latent field |
| 4 | ui_lambda_x | 0.147 | 0.154 | 0.153 | Latent field |
| 5 | ui_lambda_x | 0.149 | 0.163 | 0.154 | Latent field |
| 6 | ui_lambda_x | 0.155 | 0.137 | 0.137 | Latent field |
| 7 | ui_lambda_x | 0.149 | 0.154 | 0.144 | Latent field |
| 8 | ui_lambda_x | 0.123 | 0.134 | 0.133 | Latent field |
| 9 | ui_lambda_x | 0.164 | 0.158 | 0.184 | Latent field |
| 10 | ui_lambda_x | 0.240 | 0.246 | 0.233 | Latent field |
| 11 | ui_lambda_x | 0.166 | 0.163 | 0.161 | Latent field |
| 12 | ui_lambda_x | 0.172 | 0.179 | 0.172 | Latent field |
| 13 | ui_lambda_x | 0.180 | 0.168 | 0.191 | Latent field |
| 14 | ui_lambda_x | 0.129 | 0.139 | 0.134 | Latent field |
| 15 | ui_lambda_x | 0.180 | 0.175 | 0.175 | Latent field |
| 16 | ui_lambda_x | 0.155 | 0.156 | 0.158 | Latent field |
| 17 | ui_lambda_x | 0.183 | 0.175 | 0.181 | Latent field |
| 18 | ui_lambda_x | 0.163 | 0.154 | 0.137 | Latent field |
| 19 | ui_lambda_x | 0.172 | 0.154 | 0.152 | Latent field |
| 20 | ui_lambda_x | 0.151 | 0.143 | 0.141 | Latent field |
| 21 | ui_lambda_x | 0.168 | 0.155 | 0.167 | Latent field |
| 22 | ui_lambda_x | 0.157 | 0.127 | 0.134 | Latent field |
| 23 | ui_lambda_x | 0.165 | 0.155 | 0.162 | Latent field |
| 24 | ui_lambda_x | 0.157 | 0.157 | 0.165 | Latent field |
| 25 | ui_lambda_x | 0.156 | 0.178 | 0.166 | Latent field |
| 26 | ui_lambda_x | 0.174 | 0.178 | 0.174 | Latent field |
| 27 | ui_lambda_x | 0.137 | 0.142 | 0.141 | Latent field |
| 28 | ui_lambda_x | 0.149 | 0.141 | 0.150 | Latent field |
| 29 | ui_lambda_x | 0.134 | 0.136 | 0.135 | Latent field |
| 30 | ui_lambda_x | 0.140 | 0.134 | 0.143 | Latent field |
| 31 | ui_lambda_x | 0.145 | 0.132 | 0.141 | Latent field |
| 32 | ui_lambda_x | 0.157 | 0.152 | 0.164 | Latent field |
| 1 | log_sigma_ancrho_x | 0.504 | 0.504 | 0.024 | Hyper |
| 1 | log_sigma_ancalpha_x | 0.520 | 0.520 | 0.083 | Hyper |
| 1 | ui_anc_rho_x | 0.037 | 0.079 | 0.080 | Latent field |
| 2 | ui_anc_rho_x | 0.020 | 0.025 | 0.037 | Latent field |
| 3 | ui_anc_rho_x | 0.032 | 0.055 | 0.052 | Latent field |
| 4 | ui_anc_rho_x | 0.037 | 0.075 | 0.087 | Latent field |
| 5 | ui_anc_rho_x | 0.019 | 0.031 | 0.038 | Latent field |
| 6 | ui_anc_rho_x | 0.081 | 0.040 | 0.060 | Latent field |
| 7 | ui_anc_rho_x | 0.034 | 0.063 | 0.058 | Latent field |
| 8 | ui_anc_rho_x | 0.070 | 0.043 | 0.043 | Latent field |
| 9 | ui_anc_rho_x | 0.033 | 0.044 | 0.041 | Latent field |
| 10 | ui_anc_rho_x | 0.027 | 0.121 | 0.123 | Latent field |
| 11 | ui_anc_rho_x | 0.015 | 0.063 | 0.072 | Latent field |
| 12 | ui_anc_rho_x | 0.051 | 0.034 | 0.040 | Latent field |
| 13 | ui_anc_rho_x | 0.020 | 0.017 | 0.032 | Latent field |
| 14 | ui_anc_rho_x | 0.042 | 0.058 | 0.042 | Latent field |
| 15 | ui_anc_rho_x | 0.022 | 0.073 | 0.055 | Latent field |
| 16 | ui_anc_rho_x | 0.025 | 0.016 | 0.021 | Latent field |
| 17 | ui_anc_rho_x | 0.025 | 0.026 | 0.045 | Latent field |
| 18 | ui_anc_rho_x | 0.033 | 0.029 | 0.025 | Latent field |
| 19 | ui_anc_rho_x | 0.046 | 0.068 | 0.072 | Latent field |
| 20 | ui_anc_rho_x | 0.021 | 0.036 | 0.026 | Latent field |
| 21 | ui_anc_rho_x | 0.031 | 0.030 | 0.034 | Latent field |
| 22 | ui_anc_rho_x | 0.077 | 0.066 | 0.074 | Latent field |
| 23 | ui_anc_rho_x | 0.018 | 0.020 | 0.047 | Latent field |
| 24 | ui_anc_rho_x | 0.055 | 0.095 | 0.109 | Latent field |
| 25 | ui_anc_rho_x | 0.030 | 0.084 | 0.059 | Latent field |
| 26 | ui_anc_rho_x | 0.024 | 0.053 | 0.053 | Latent field |
| 27 | ui_anc_rho_x | 0.068 | 0.073 | 0.030 | Latent field |
| 28 | ui_anc_rho_x | 0.026 | 0.020 | 0.023 | Latent field |
| 29 | ui_anc_rho_x | 0.076 | 0.038 | 0.029 | Latent field |
| 30 | ui_anc_rho_x | 0.026 | 0.024 | 0.022 | Latent field |
| 31 | ui_anc_rho_x | 0.027 | 0.041 | 0.052 | Latent field |
| 32 | ui_anc_rho_x | 0.028 | 0.019 | 0.024 | Latent field |
| 1 | ui_anc_alpha_x | 0.033 | 0.070 | 0.076 | Latent field |
| 2 | ui_anc_alpha_x | 0.045 | 0.052 | 0.072 | Latent field |
| 3 | ui_anc_alpha_x | 0.079 | 0.081 | 0.088 | Latent field |
| 4 | ui_anc_alpha_x | 0.060 | 0.058 | 0.061 | Latent field |
| 5 | ui_anc_alpha_x | 0.032 | 0.037 | 0.035 | Latent field |
| 6 | ui_anc_alpha_x | 0.028 | 0.019 | 0.034 | Latent field |
| 7 | ui_anc_alpha_x | 0.060 | 0.058 | 0.070 | Latent field |
| 8 | ui_anc_alpha_x | 0.052 | 0.059 | 0.052 | Latent field |
| 9 | ui_anc_alpha_x | 0.059 | 0.082 | 0.070 | Latent field |
| 10 | ui_anc_alpha_x | 0.021 | 0.066 | 0.058 | Latent field |
| 11 | ui_anc_alpha_x | 0.031 | 0.029 | 0.033 | Latent field |
| 12 | ui_anc_alpha_x | 0.048 | 0.063 | 0.059 | Latent field |
| 13 | ui_anc_alpha_x | 0.049 | 0.043 | 0.053 | Latent field |
| 14 | ui_anc_alpha_x | 0.048 | 0.060 | 0.068 | Latent field |
| 15 | ui_anc_alpha_x | 0.068 | 0.074 | 0.073 | Latent field |
| 16 | ui_anc_alpha_x | 0.094 | 0.098 | 0.033 | Latent field |
| 17 | ui_anc_alpha_x | 0.034 | 0.036 | 0.047 | Latent field |
| 18 | ui_anc_alpha_x | 0.138 | 0.135 | 0.053 | Latent field |
| 19 | ui_anc_alpha_x | 0.052 | 0.046 | 0.050 | Latent field |
| 20 | ui_anc_alpha_x | 0.086 | 0.070 | 0.071 | Latent field |
| 21 | ui_anc_alpha_x | 0.075 | 0.055 | 0.048 | Latent field |
| 22 | ui_anc_alpha_x | 0.084 | 0.053 | 0.039 | Latent field |
| 23 | ui_anc_alpha_x | 0.044 | 0.048 | 0.046 | Latent field |
| 24 | ui_anc_alpha_x | 0.055 | 0.068 | 0.049 | Latent field |
| 25 | ui_anc_alpha_x | 0.110 | 0.081 | 0.087 | Latent field |
| 26 | ui_anc_alpha_x | 0.054 | 0.054 | 0.059 | Latent field |
| 27 | ui_anc_alpha_x | 0.041 | 0.057 | 0.062 | Latent field |
| 28 | ui_anc_alpha_x | 0.034 | 0.036 | 0.028 | Latent field |
| 29 | ui_anc_alpha_x | 0.133 | 0.120 | 0.042 | Latent field |
| 30 | ui_anc_alpha_x | 0.056 | 0.063 | 0.051 | Latent field |
| 31 | ui_anc_alpha_x | 0.030 | 0.049 | 0.040 | Latent field |
| 32 | ui_anc_alpha_x | 0.062 | 0.053 | 0.040 | Latent field |
| 1 | log_sigma_or_gamma | 0.524 | 0.524 | 0.048 | Hyper |
| 1 | log_or_gamma | 0.022 | 0.110 | 0.115 | Latent field |
| 2 | log_or_gamma | 0.027 | 0.088 | 0.084 | Latent field |
| 3 | log_or_gamma | 0.033 | 0.042 | 0.028 | Latent field |
| 4 | log_or_gamma | 0.042 | 0.085 | 0.076 | Latent field |
| 5 | log_or_gamma | 0.044 | 0.105 | 0.101 | Latent field |
| 6 | log_or_gamma | 0.056 | 0.094 | 0.092 | Latent field |
| 7 | log_or_gamma | 0.022 | 0.023 | 0.019 | Latent field |
| 8 | log_or_gamma | 0.044 | 0.038 | 0.043 | Latent field |
| 9 | log_or_gamma | 0.046 | 0.065 | 0.065 | Latent field |
| 10 | log_or_gamma | 0.017 | 0.027 | 0.039 | Latent field |
| 11 | log_or_gamma | 0.013 | 0.023 | 0.032 | Latent field |
| 12 | log_or_gamma | 0.018 | 0.107 | 0.110 | Latent field |
| 13 | log_or_gamma | 0.020 | 0.060 | 0.076 | Latent field |
| 14 | log_or_gamma | 0.025 | 0.024 | 0.035 | Latent field |
| 15 | log_or_gamma | 0.035 | 0.035 | 0.035 | Latent field |
| 16 | log_or_gamma | 0.041 | 0.026 | 0.029 | Latent field |
| 17 | log_or_gamma | 0.033 | 0.036 | 0.039 | Latent field |
| 18 | log_or_gamma | 0.041 | 0.137 | 0.144 | Latent field |
| 19 | log_or_gamma | 0.027 | 0.037 | 0.045 | Latent field |
| 20 | log_or_gamma | 0.018 | 0.029 | 0.038 | Latent field |
| 21 | log_or_gamma | 0.049 | 0.053 | 0.053 | Latent field |
| 22 | log_or_gamma | 0.026 | 0.022 | 0.030 | Latent field |
| 23 | log_or_gamma | 0.023 | 0.029 | 0.026 | Latent field |
| 24 | log_or_gamma | 0.015 | 0.035 | 0.040 | Latent field |
| 25 | log_or_gamma | 0.052 | 0.055 | 0.046 | Latent field |
| 26 | log_or_gamma | 0.029 | 0.048 | 0.057 | Latent field |
| 27 | log_or_gamma | 0.024 | 0.016 | 0.028 | Latent field |
| 28 | log_or_gamma | 0.051 | 0.036 | 0.053 | Latent field |
| 29 | log_or_gamma | 0.021 | 0.052 | 0.050 | Latent field |
| 30 | log_or_gamma | 0.050 | 0.075 | 0.074 | Latent field |
| 31 | log_or_gamma | 0.037 | 0.060 | 0.043 | Latent field |
| 32 | log_or_gamma | 0.017 | 0.041 | 0.055 | Latent field |
This plot could be improved by making the red to green transition smooth, then adding legend for KS difference. The density plots could be filled according to KS difference.
summary_ks_plot <- function(ks_summary, method1, method2) {
ks_method1 <- paste0("KS(", method1, ", tmbstan)")
ks_method2 <- paste0("KS(", method2, ", tmbstan)")
xy_length <- min(1, max(ks_summary[[ks_method1]], ks_summary[[ks_method2]]) + 0.05)
x <- y <- seq(0, xy_length, length.out = 20)
gradient_base <- expand.grid(x, y) %>%
mutate(diff = Var1 - Var2) %>%
ggplot(aes(x = Var1, y = Var2)) +
geom_tile(aes(fill = diff), alpha = 0.7) +
scale_fill_gradientn(
colours = c("#56B4E9", "white", "#009E73"),
rescaler = ~ scales::rescale_mid(.x, mid = 0)
)
densityplot <- gradient_base +
stat_density_2d(data = ks_summary, aes(x = .data[[ks_method1]], y = .data[[ks_method2]], linetype = Type), col = "black") +
xlim(0, xy_length) +
ylim(0, xy_length) +
geom_abline(intercept = 0, slope = 1, linetype = "dashed", alpha = 0.5) +
scale_linetype_manual(values = c("solid", "dashed")) +
labs(x = ks_method1, y = ks_method2) +
theme_minimal() +
guides(fill = "none") +
theme(
legend.position = "bottom"
)
ks_summary[["KS difference"]] <- ks_summary[[ks_method1]] - ks_summary[[ks_method2]]
ridgeplot <- ggplot(ks_summary, aes(y = Type, x = `KS difference`)) +
ggridges::geom_density_ridges(alpha = 0.7, fill = NA, aes(linetype = Type)) +
coord_flip() +
scale_linetype_manual(values = c("solid", "dashed")) +
labs(y = "", x = paste0(ks_method1, " - ", ks_method2)) +
guides(linetype = FALSE) +
theme_minimal()
densityplot + ridgeplot
}
summary_ks_plot(ks_summary, "TMB", "aghq")
## Warning: Removed 76 rows containing missing values (`geom_tile()`).
## Picking joint bandwidth of 0.00982
summary_ks_plot(ks_summary, "TMB", "adam")
## Warning: Removed 76 rows containing missing values (`geom_tile()`).
## Picking joint bandwidth of 0.0103
summary_ks_plot(ks_summary, "aghq", "adam")
## Warning: Removed 76 rows containing missing values (`geom_tile()`).
## Picking joint bandwidth of 0.24
(ks_summary_out <- ks_summary %>%
group_by(Type) %>%
summarise(
TMB = mean(`KS(TMB, tmbstan)`),
aghq = mean(`KS(aghq, tmbstan)`),
adam = mean(`KS(adam, tmbstan)`)
))
Compute \(p\)-values, then plot expecting \(\mathcal{U}(0, 1)\). If not conclude there is a significant difference.
#' To write!
Save some things for output:
pdf("ks_summary_tmb_adam.pdf", h = 4, w = 6.25)
summary_ks_plot(ks_summary, "TMB", "adam")
## Warning: Removed 76 rows containing missing values (`geom_tile()`).
## Picking joint bandwidth of 0.0103
dev.off()
## quartz_off_screen
## 2
saveRDS(ks_summary_out, "ks_summary_out.rds")
The following parameters have KS values greater than 0.5 in both dimensions.
(big_ks <- ks_summary %>%
filter(`KS(aghq, tmbstan)` + `KS(TMB, tmbstan)` > 0.5) %>%
pull(Parameter))
## [1] "logit_phi_rho_x" "log_sigma_rho_x" "logit_phi_rho_xs" "log_sigma_rho_xs" "logit_phi_rho_a"
## [6] "log_sigma_rho_a" "logit_phi_rho_as" "log_sigma_rho_as" "log_sigma_rho_xa" "logit_phi_alpha_x"
## [11] "log_sigma_alpha_x" "logit_phi_alpha_xs" "log_sigma_alpha_xs" "logit_phi_alpha_a" "log_sigma_alpha_a"
## [16] "logit_phi_alpha_as" "log_sigma_alpha_as" "log_sigma_alpha_xa" "OmegaT_raw" "log_betaT"
## [21] "log_sigma_lambda_x" "log_sigma_ancrho_x" "log_sigma_ancalpha_x" "log_sigma_or_gamma"
#' To write!
Suppose we have two sets of samples from the posterior.
For each sample we are going to want to evaluate the log-likelihood, so that we can calculate the log-likelihood ratios.
We can extract the TMB objective function for the log-likelihood as follows:
tmb$fit$obj$fn()
## [1] NaN
#' To write!